Find least Square regression

Question

Find the least squares regression line for the data points (1,1),(2,3),(4,5)

Answer 1

The first thing we must find is the slope, β₁

 

The equation for β₁ = SS_xy/SS_xx = Σ(X_i - X-bar)(Y_i - Y-bar)/∑(X_i - X-bar)²

 

X-bar, the mean of the X-coordinates, is (1+2+4)/3 = 2.3333

Y-bar, the mean of the Y-coordinates, is (1+3+5)/3 = 3

 

Our numerator summation therefore is:

(1 - 2.3333)(1-3) + (2 - 2.3333)(3-3) + (4 - 2.3333)(5 - 3) = 2.6666 + 0 + 3.3334 = 6

 

Our denominator summation is:

(1 - 2.3333)² + (2 - 2.3333)² + (4 - 2.3333)² = 1.7777 + .1111 + 2.7777 = 4.6666

 

So β1 = 6/4.6666 = 1.2857

 

To find the intercept, β₀ = Y-bar - β1 * X-bar = 3 - 1.2857(2.3333) = 3 - 3 = 0

 

So the regression equation should be Y = 1.2857X

 

When we only have a few points it's not hard to do by hand.  I would suggest using matrices or statistical software should the number of observations rise.

 

 

 

Answer 2

Do you have access to any graphing technology, or are you expected to do this entirely by hand with matrices?